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Theorem signslema 28392
Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
Hypotheses
Ref Expression
signslema.1  |-  ( ph  ->  E  e.  NN0 )
signslema.2  |-  ( ph  ->  F  e.  NN0 )
signslema.3  |-  ( ph  ->  G  e.  NN0 )
signslema.4  |-  ( ph  ->  H  e.  NN0 )
signslema.5  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
signslema.6  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
Assertion
Ref Expression
signslema  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )

Proof of Theorem signslema
StepHypRef Expression
1 signslema.5 . . . . . 6  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
21simpld 459 . . . . 5  |-  ( ph  ->  E  <  G )
32adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  E  <  G )
4 signslema.4 . . . . . . . . . 10  |-  ( ph  ->  H  e.  NN0 )
54nn0cnd 10860 . . . . . . . . 9  |-  ( ph  ->  H  e.  CC )
6 signslema.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  NN0 )
76nn0cnd 10860 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
85, 7subcld 9936 . . . . . . . 8  |-  ( ph  ->  ( H  -  F
)  e.  CC )
9 signslema.3 . . . . . . . . . 10  |-  ( ph  ->  G  e.  NN0 )
109nn0cnd 10860 . . . . . . . . 9  |-  ( ph  ->  G  e.  CC )
11 signslema.1 . . . . . . . . . 10  |-  ( ph  ->  E  e.  NN0 )
1211nn0cnd 10860 . . . . . . . . 9  |-  ( ph  ->  E  e.  CC )
1310, 12subcld 9936 . . . . . . . 8  |-  ( ph  ->  ( G  -  E
)  e.  CC )
148, 13subeq0ad 9946 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  <-> 
( H  -  F
)  =  ( G  -  E ) ) )
1514biimpa 484 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( H  -  F )  =  ( G  -  E ) )
1615breq2d 4449 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
0  <  ( H  -  F )  <->  0  <  ( G  -  E ) ) )
176nn0red 10859 . . . . . . 7  |-  ( ph  ->  F  e.  RR )
184nn0red 10859 . . . . . . 7  |-  ( ph  ->  H  e.  RR )
1917, 18posdifd 10145 . . . . . 6  |-  ( ph  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2111nn0red 10859 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
229nn0red 10859 . . . . . . 7  |-  ( ph  ->  G  e.  RR )
2321, 22posdifd 10145 . . . . . 6  |-  ( ph  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2516, 20, 243bitr4rd 286 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  F  <  H ) )
263, 25mpbid 210 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  F  <  H )
27 0red 9600 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  e.  RR )
2822, 21resubcld 9993 . . . . . 6  |-  ( ph  ->  ( G  -  E
)  e.  RR )
2928adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  e.  RR )
3018, 17resubcld 9993 . . . . . 6  |-  ( ph  ->  ( H  -  F
)  e.  RR )
3130adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( H  -  F )  e.  RR )
322adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  E  <  G )
3323adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
3432, 33mpbid 210 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( G  -  E
) )
35 2pos 10633 . . . . . . 7  |-  0  <  2
36 breq2 4441 . . . . . . 7  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  (
0  <  ( ( H  -  F )  -  ( G  -  E ) )  <->  0  <  2 ) )
3735, 36mpbiri 233 . . . . . 6  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
3828, 30posdifd 10145 . . . . . . 7  |-  ( ph  ->  ( ( G  -  E )  <  ( H  -  F )  <->  0  <  ( ( H  -  F )  -  ( G  -  E
) ) ) )
3938biimpar 485 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( H  -  F
)  -  ( G  -  E ) ) )  ->  ( G  -  E )  <  ( H  -  F )
)
4037, 39sylan2 474 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  <  ( H  -  F
) )
4127, 29, 31, 34, 40lttrd 9746 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( H  -  F
) )
4219adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
4341, 42mpbird 232 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  F  <  H )
445, 10, 7, 12sub4d 9985 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  =  ( ( H  -  F )  -  ( G  -  E ) ) )
45 signslema.6 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
4644, 45eqeltrrd 2532 . . . 4  |-  ( ph  ->  ( ( H  -  F )  -  ( G  -  E )
)  e.  { 0 ,  2 } )
47 ovex 6309 . . . . 5  |-  ( ( H  -  F )  -  ( G  -  E ) )  e. 
_V
4847elpr 4032 . . . 4  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  e.  { 0 ,  2 }  <->  ( (
( H  -  F
)  -  ( G  -  E ) )  =  0  \/  (
( H  -  F
)  -  ( G  -  E ) )  =  2 ) )
4946, 48sylib 196 . . 3  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  \/  ( ( H  -  F )  -  ( G  -  E
) )  =  2 ) )
5026, 43, 49mpjaodan 786 . 2  |-  ( ph  ->  F  <  H )
511simprd 463 . . . . 5  |-  ( ph  ->  -.  2  ||  ( G  -  E )
)
5251adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( G  -  E ) )
5315breq2d 4449 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
2  ||  ( H  -  F )  <->  2  ||  ( G  -  E
) ) )
5452, 53mtbird 301 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( H  -  F ) )
55 2z 10902 . . . . . . 7  |-  2  e.  ZZ
569nn0zd 10972 . . . . . . . 8  |-  ( ph  ->  G  e.  ZZ )
5711nn0zd 10972 . . . . . . . 8  |-  ( ph  ->  E  e.  ZZ )
5856, 57zsubcld 10979 . . . . . . 7  |-  ( ph  ->  ( G  -  E
)  e.  ZZ )
59 dvdsaddr 13902 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  ( G  -  E
)  e.  ZZ )  ->  ( 2  ||  ( G  -  E
)  <->  2  ||  (
( G  -  E
)  +  2 ) ) )
6055, 58, 59sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  ||  ( G  -  E )  <->  2 
||  ( ( G  -  E )  +  2 ) ) )
6151, 60mtbid 300 . . . . 5  |-  ( ph  ->  -.  2  ||  (
( G  -  E
)  +  2 ) )
6261adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( ( G  -  E )  +  2 ) )
63 2cnd 10614 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
648, 13, 63subaddd 9954 . . . . . 6  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  2  <-> 
( ( G  -  E )  +  2 )  =  ( H  -  F ) ) )
6564biimpa 484 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
( G  -  E
)  +  2 )  =  ( H  -  F ) )
6665breq2d 4449 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
2  ||  ( ( G  -  E )  +  2 )  <->  2  ||  ( H  -  F
) ) )
6762, 66mtbid 300 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( H  -  F ) )
6854, 67, 49mpjaodan 786 . 2  |-  ( ph  ->  -.  2  ||  ( H  -  F )
)
6950, 68jca 532 1  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   {cpr 4016   class class class wbr 4437  (class class class)co 6281   RRcr 9494   0cc0 9495    + caddc 9498    < clt 9631    - cmin 9810   2c2 10591   NN0cn0 10801   ZZcz 10870    || cdvds 13863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-dvds 13864
This theorem is referenced by: (None)
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